Question

a bacteria culture is known to grow at a rate proportional to the amount present. after one hour, 1000 strands of the bacteria are observed in the culture; and after four hours, 30000 strands. Find an expression fo the approximate number of strands of the bacteria present in the culture at any time t.

Answer 1

Let $P(t)$ be the size of the population of the bacteria culture at any time $t.$

Then

$$\dfrac{dP}{dt}=kP$$

$$\dfrac{dP}{P}=kdt$$

$$P(t)=P(0)e^{kt}$$

Given $P(1)=1000, P(4)=30000$

$$\dfrac{P(4)}{P(1)}=\dfrac{P(0)e^{4k}}{P(0)e^{k}}=\dfrac{30000}{1000}$$

$$e^{3k}=30$$

$$k=\dfrac{\ln 30}{3}$$

$$P(t)=P(0)e^{(\ln 30/3)t}$$

$$P(t)=P(0)(30)^{t/3}$$

$$P(1)=P(0)(30)^{1/3}=1000$$

$$P(0)=\dfrac{1000}{\sqrt[3]{30}}$$

$$P(t)=\dfrac{1000}{\sqrt[3]{30}}(30)^{t/3}$$

$$P(t)\approx322(30)^{t/3}, t\geq 0$$